Clustering and classification

Data wrangling exercises are done in the corresponding R script.

Analysis exercises

Firstly, the Boston dataset is loaded from the MASS package:

library(MASS)
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following object is masked from 'package:MASS':
## 
##     select
## The following objects are masked from 'package:stats':
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##     filter, lag
## The following objects are masked from 'package:base':
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##     intersect, setdiff, setequal, union
data(Boston)
str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
dim(Boston)
## [1] 506  14

The data has 14 variables, all of them are either numeric or integer. Variables include e.g. crime rate by town per capita (crim), nitrogen oxides concentration (nox), index of accessibility to radial highways (rad), pupil-teacher ratio by town (ptratio), average number of rooms per dwelling (rm) and others.

summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

rad (index of accessibility ro radial highways) varies from 1 to 24. lstat (lower status of the population (in percents)) has several outliers: min lstat is 1.73%, max lstat is 37.97 %, and the mean is 12.65%. There is also a noticeable outlier in crim variable: max crim = 87.98. An outlier is also present in black variable (1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town): min black = 0.32.

pairs(Boston)

There is a clear hyperbolic relationship between nox and dis (weighted mean of distances to five Boston employment centres) variables: the more the concentration of nitrogen oxides, the less the weighted mean of distances to employment centers. Thus, we can say that there is a correlation between nitrogen oxides and employments centers.

There is also a hyperbolic relationship between lstat and medv (median value of owner-occupied homes in $1000s) variables: the lower the status of the population, the less the median cost of the homes in the area, which is pretty logical.

There is almost a linear correlation between rm (average number of rooms per dwelling) and lstat variables: the more rooms in the dwelling, the more the median cost of the homes and vice versa.

Now the dataset will be standardized and a new variable will be added to the dataset (the old variable crim will be dropped):

boston_scaled <- scale(Boston)
summary(boston_scaled)
##       crim                 zn               indus              chas        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563   Min.   :-0.2723  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668   1st Qu.:-0.2723  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109   Median :-0.2723  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150   3rd Qu.:-0.2723  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202   Max.   : 3.6648  
##       nox                rm               age               dis         
##  Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331   Min.   :-1.2658  
##  1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366   1st Qu.:-0.8049  
##  Median :-0.1441   Median :-0.1084   Median : 0.3171   Median :-0.2790  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059   3rd Qu.: 0.6617  
##  Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164   Max.   : 3.9566  
##       rad               tax             ptratio            black        
##  Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047   Min.   :-3.9033  
##  1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876   1st Qu.: 0.2049  
##  Median :-0.5225   Median :-0.4642   Median : 0.2746   Median : 0.3808  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.4332  
##  Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372   Max.   : 0.4406  
##      lstat              medv        
##  Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 3.5453   Max.   : 2.9865
boston_scaled <- as.data.frame(boston_scaled)
summary(boston_scaled$crim)
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -0.419367 -0.410563 -0.390280  0.000000  0.007389  9.924110
#creating a new variable
breakpoints <- quantile(boston_scaled$crim)
labels <- c('low','med_low','med_high','high')
crime <- cut(boston_scaled$crim, breaks = breakpoints, include.lowest = TRUE, label=labels)

#dropping and adding variables
boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled <- data.frame(boston_scaled, crime)

Diving the data to train and test data:

n <- nrow(boston_scaled)
random_rows <- sample(n,  size = n * 0.8)
train_data <- boston_scaled[random_rows,]
test_data <- boston_scaled[-random_rows,]

Linear discriminant analysis

Fitting and plotting the LDA:

lda.fit <- lda(crime ~ ., data = train_data)
classes <- as.numeric(train_data$crime)
plot(lda.fit, dimen = 2,col=classes,pch=classes)

Predicting the classes with the LDA model:

correct_classes <- test_data[,"crime"]
test_data <- dplyr::select(test_data, -crime)

lda.pred <- predict(lda.fit, newdata = test_data)
table(correct = correct_classes, predicted = lda.pred$class)
##           predicted
## correct    low med_low med_high high
##   low       14       5        2    0
##   med_low   10      17        6    0
##   med_high   1       9       11    1
##   high       0       0        0   26

Overall results show that high crime rate was predicted correctly (there is only one case when medium high crime rate was predicted wrong as high). Low crime rate was predicted correctly for 17 cases, for 13 cases it was predicted as med_low and for 2 - as med_high. Medium low crime rate was predicted correctly 15 times with only 6 errors.

Calculating the distances and visualizing the clusters:

library(MASS)
data('Boston')
boston_scaled_again <- scale(Boston)

dist_eu <- dist(boston_scaled_again)   

km <-kmeans(Boston, centers = 2)
pairs(Boston, col = km$cluster)

The optimal number of clusters is 2: 3 look good already, but one of them (the black one) doesn’t seem to be of great significance. More than 3 clusters is abundant. In case of rad, tax and ptratio the red cluster clearly shows outliers. There are overlapping clusters in lstat and medv plot: one is corresponding to the bigger amount of people who are of lower status and to the smaller median cost of homes, and the other is corresponding to the bigger amount of people with higher status and to bigger median cost of homes (basically, it is poverty vs richness dichotomy)

Bonus Task:

model_predictors <- dplyr::select(train_data, -crime)
# check the dimensions
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)

install.packages("plotly")
## Installing package into '/home/aleksandrako/R/x86_64-pc-linux-gnu-library/3.6'
## (as 'lib' is unspecified)
library("plotly")
## Loading required package: ggplot2
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## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
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##     last_plot
## The following object is masked from 'package:MASS':
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##     layout
#colors from the crime classes
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers',color=train_data$crime)